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HMMT 二月 2023 · 冲刺赛 · 第 3 题

HMMT February 2023 — Guts Round — Problem 3

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [10] Let ABCD be a convex quadrilateral such that ∠ ABD = ∠ BCD = 90 , and let M be the midpoint of segment BD . Suppose that CM = 2 and AM = 3. Compute AD .
解析
  1. [10] Let ABCD be a convex quadrilateral such that ∠ ABD = ∠ BCD = 90 , and let M be the midpoint of segment BD . Suppose that CM = 2 and AM = 3. Compute AD . Proposed by: Luke Robitaille, Milan Haiman √ Answer: 21 Solution: Since triangle BCD is a right triangle, we have CM = BM = DM = 2. With AM = 3 √ ◦ and ∠ ABM = 90 , we get AB = 5. Now 2 2 2 AD = AB + BD = 5 + 16 = 21 , √ so AD = 21.